Oran Mcgall wrote
On Thu, 24 Jun 2010, Oran Magal wrote:
>> What properties are such that two structures can be isomorphic, and
> yet differ with regard to these properties? Informally speaking,
> wouldn't those be: those properties that aren't structural properties?
> Those properties would then have to be properties specific to an
> _interpretation_ of the structure; something to do with their
> 'standard' interpretation, perhaps, such as the natural numbers for PA
> or the 'intuitive' objects point, line, plane for the axioms of
> Euclidean geometry.
>> But then, these are properties that are not given in any formalism,
> but rather given in the usual concepts or associations or 'intuitions'
> standardly connected with the structure in question. And given that
> these are properties of a specific _interpretation_, how could any
> other isomorphic interpretation of the same structure be identical
> with respect to them without being the very same interpretation, and
> therefore trivially identical to it?
>> What I have been trying to say, perhaps not too articulately, is that
> I'm not sure what mathematical properties, which are not already
> captured by isomorphism, you have in mind with respect to which the
> question of identity/difference arises.
>
There are problems about structures in a fixed vocabulary that are
proved using the properties of a representation of these structures in a
(larger) vocabulary.
For example Whitehead's problem about groups in the vocabulary (+,=) was
solved by Shelah considering the structure to have universe aleph_1 - thus
in the larger vocabulary with the epsilon symbol. There are many other
examples of this phenomena.
John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607